What’s the problem with teachers’ mathematical knowledge and whose problem is it?

One might think that success in school as a child is adequate mathematical preparation for K-8 teaching. Evidence, however, suggests that success in school does not guarantee that one knows a subject well. More to the point, such success does not necessarily mean a teacher knows the subject well enough to help others learn it. While most teachers liked school as children and were relatively successful, many lack important mathematical knowledge needed for effective teaching.

The mathematical demands of K-8 teaching are quite substantial. In addition to knowing the mathematical definitions of terms, teachers must be able to use definitions effectively when teaching. In defining a term, they need to be able to find language that is meaningful to children, yet mathematically correct. Defining an even number to be an integer multiple of 2 is of little use if students do not know the terms integer or multiple. Defining an even number to be a whole number that is two times another number, while perhaps accessible to children, admits all whole numbers (e.g., 1 is two times one half). Amending this definition to say, two times another whole number, may seem to solve the problem, but this definition excludes negative multiples of 2. Later, when students learn about negative integers, they should not have to unlearn an earlier definition. Changing the definition to a whole number is even if it is 2 times another whole number is mathematically honest, yet respects the scope of children’s experience.

These subtle and mathematically demanding issues would challenge most college students, even mathematics majors. Classroom teachers, however, contend regularly with mathematical problems of exactly this type. Teachers must routinely: give clear explanations, choose useful examples, evaluate students’ ideas, select appropriate representations, modify problems to be easier or harder, recognize different ways to solve the same problem, explain goals and mathematical purposes to others, and build correspondences between models and procedures.

These are relatively uncommon skills in our society and are rarely taught. They require significant mathematical knowledge not typically needed by people who do not teach. Yet teachers need to be proficient at these tasks amidst the busy flow of classroom life. For instance, anyone who knows the content should be able to determine that a pupil has produced a wrong answer to a problem. But figuring out what a student did wrong—spotting the method and guessing its rationale—requires greater skill and knowledge. Likewise, teachers must recognize when a “right” answer is the result of faulty thinking. These analyses represent mathematically demanding activities specifically required of those who teach. Mathematics of this kind is mathematical knowledge for teaching. As such, it is unique to the profession of teaching and is distinct as a body of knowledge within the field of mathematics.

But where and when do teachers learn such mathematics?